Geodesic
Solitary wave effects of Woods--Saxon potential in Schr\"odinger equation with 3d cubic nonlinearity
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 17 (2024) no. 2, pp. 68-82

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In this research article, we apply the generalized projective Riccati equation method to construct traveling wave solutions of the 3d cubic focusing nonlinear Schrödinger equation with Woods–Saxon potential. The generalized projective Riccati equation method is a powerful and effective mathematical tool for obtaining exact solutions of nonlinear partial differential equations, and it allows us to derive a variety of traveling wave solutions of the 3d cubic focusing nonlinear Schrödinger equation with Woods–Saxon potential. These solutions contain periodic wave solutions, bright and dark soliton solutions. The study of many physical systems, such as Bose–Einstein condensates and nonlinear optics, that give rise to the nonlinear Schrödinger equation. We provide a detailed description of the generalized projective Riccati equation method in the paper, and demonstrate its usefulness in solving the nonlinear Schrödinger equation with Woods–Saxon potential. We present various graphical representations of the obtained solutions using MATLAB software, and analyze their characteristics. Our results provide new insights into the behavior of the 3d cubic focusing nonlinear Schrödinger equation with Woods–Saxon potential, and have potential applications in numerous fields of physics, as well as nonlinear optics and condensed matter physics.
Keywords: 3d cubic focusing nonlinear Schrödinger equation, Woods–Saxon potential, traveling wave solution, generalized projective Riccati equation method (GPREM). 
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     author = {Mustafa Inc and Muhammad Sajid Iqbal and Ali Hasan Ali and Zuha Manzoor and Farrah Ashraf},
     title = {Solitary wave effects of {Woods--Saxon} potential in {Schr\"odinger} equation with 3d cubic nonlinearity},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
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Mustafa Inc; Muhammad Sajid Iqbal; Ali Hasan Ali; Zuha Manzoor; Farrah Ashraf. Solitary wave effects of Woods--Saxon potential in Schr\"odinger equation with 3d cubic nonlinearity. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 17 (2024) no. 2, pp. 68-82. http://geodesic.mathdoc.fr/item/VYURU_2024_17_2_a5/